On Read's Type Operators on Hilbert Spaces

Grivaux, Sophie; Roginskaya, Maria

doi:10.1093/imrn/rnn083

Cited by 21 publications

(59 citation statements)

References 16 publications

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“…In [40] it was shown that "many" Hilbert-space operators are orbit reflexive, for instance, normal, compact, algebraic operators and contractions. Examples of Hilbert-space operators that are not orbit-reflexive only recently appeared in [33,47]. It is an open question whether every power bounded operator T ∶ X → X is orbit reflexive.…”

Section: Corollary 92 Let T ∶ X → X Be An Operator If T Is Cyclic mentioning

confidence: 99%

Recurrent Linear Operators

Costakis

Manoussos

Parissis

2013

Complex Anal. Oper. Theory

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Abstract. We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions. The most studied notion in linear dynamics is that of hypercyclicity: a bounded linear operator T acting on a separable Banach space is hypercyclic if there exists a vector whose orbit under T is dense in the space. On the other hand, a very central notion in topological dynamics is that of recurrence. This notion goes back to Poincaré and Birkhoff and it refers to the existence of points in the space for which parts of their orbits under a continuous map "return" to themselves. The purpose of this note is the study of the notion of recurrence, together with its variations, in the context of linear dynamics. Some examples and characterizations of recurrence for special In an effort to characterize recurrent linear operators one many times falls back to the notion of hypercyclicity. This is for example the case when we study the recurrence properties of backwards shifts, say on ℓ 2 (Z). The reason behind is that, according to a result of Seceleanu, [51], the orbits of these operators satisfy a zero-one law: if the orbit of a weighted backward shift contains a non-zero limit point then the corresponding shift is actually hypercyclic. Thus a weighted backward shift on ℓ 2 (Z) is recurrent if and only if it is hypercyclic. The same equivalence is true, albeit for different reasons, for the adjoint of a multiplication operators on the Hardy space H 2 (D). These connections to hypercyclicity, already observed in [17], come up naturally and thus motivate a further search on whether the properties of recurrent operators resemble the properties of hypercyclic ones, in general. It turns out that, indeed, there are many structural similarities between the set of hypercyclic vectors and the set of recurrent vectors in the sense that they exhibit the same invariances. Furthermore, the spectral properties of hypercyclic and recurrent operators are somewhat similar, although this vague statement should be interpreted with some care. However, these similarities cannot be pushed too much as there are obviously many classes of operators which are recurrent without being hypercyclic. One can find such examples among composition operators on the Hardy space H 2 (D). However, the primordial example is given just by considering unimodular multiples of the identity operator. A more general class for which one needs to address the recurrence properties independently of hypercyclicity is that of unitary operators on Hilbert spaces.The discussion above hopefully justifies why we will shortly recall a full set of definitions rel...

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Section: Corollary 92 Let T ∶ X → X Be An Operator If T Is Cyclic mentioning

confidence: 99%

Recurrent Linear Operators

Costakis

Manoussos

Parissis

2013

Complex Anal. Oper. Theory

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“…In particular, in a profound study concerning spaces admitting operators without non-trivial invariant subspaces, Read has proved that every separable Banach space that contains either c 0 or a complemented subspace isomorphic to 1 or J ∞ , admits an operator without non-trivial closed invariant subspaces [30]. A comprehensive study of Read's methods of constructing operators with no non-trivial invariant subspaces can be found in [19,20]. Also recently a non-reflexive hereditarily indecomposable (HI) Banach space X K with the 'scalar plus compact' property has been constructed [7].…”

Section: Introductionmentioning

confidence: 99%

A reflexive hereditarily indecomposable space with the hereditary invariant subspace property

Argyros

Motakis

2013

Proceedings of the London Mathematical Society

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A separable Banach space X satisfies the invariant subspace property (ISP) if every bounded linear operator T ∈ L(X) admits a non-trivial closed invariant subspace. In this paper, we present the first known example of a reflexive Banach space X ISP satisfying the ISP. Moreover, this is the first known example of a Banach space satisfying the hereditary ISP, namely every infinitedimensional subspace of it satisfies the ISP. The space X ISP is hereditarily indecomposable (HI) and every operator T ∈ L(X ISP) is of the form λI + S with S a strictly singular operator. The critical property of the strictly singular operators of X ISP is that the composition of any three of them is a compact one. The construction of X ISP is based on saturation methods and it uses as an unconditional frame Tsirelson space. The new ingredient in the definition of the space is the saturation under constraints, a method initialized in a fundamental work of Edward Odell and Thomas Schlumprecht.

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“…This was shown in the paper [12], where operators on the Hilbert space with few invariant closed subsets were constructed. We quote here the main result of [12] (combined with a remark from [13]), which shows which kind of properties of Read's type operators can be enforced in the Hilbertian setting. Theorem 1.3 [12].…”

Section: Introductionmentioning

confidence: 92%

“…We quote here the main result of [12] (combined with a remark from [13]), which shows which kind of properties of Read's type operators can be enforced in the Hilbertian setting. Theorem 1.3 [12]. There exists a bounded operator on a separable (real or complex) Hilbert space H which satisfies the following two properties:…”

Section: Introductionmentioning

confidence: 98%

A general approach to Read's type constructions of operators without non-trivial invariant closed subspaces

Grivaux

Roginskaya

2014

Proceedings of the London Mathematical Society

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We present a general method for constructing operators without non-trivial invariant closed subsets on a large class of non-reflexive Banach spaces. In particular, our approach unifies and generalizes several constructions due to Read of operators without non-trivial invariant subspaces on the spaces 1, c0 or 2 J, and without non-trivial invariant subsets on 1. We also investigate how far our methods can be extended to the Hilbertian setting, and construct an operator on a quasi-reflexive dual Banach space which has no non-trivial w * -closed invariant subspace.We retrieve in particular the existence of an operator on 1 with no non-trivial invariant closed subset [22], and improve the results of [23] by showing that c 0 and 2 J support operators without non-trivial invariant closed subsets.An interesting consequence of Theorem 1.1 is the following corollary.Corollary 1.2. Let X be an infinite-dimensional non-reflexive separable space having an unconditional basis. Then there exists a bounded operator on X with no non-trivial invariant closed subset.

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On Read's Type Operators on Hilbert Spaces

Cited by 21 publications

References 16 publications

Recurrent Linear Operators

Recurrent Linear Operators

A reflexive hereditarily indecomposable space with the hereditary invariant subspace property

A general approach to Read's type constructions of operators without non-trivial invariant closed subspaces

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